Trade and Inequality in the Spatial Economy

Farid Farrokhi1 and David Jinkins∗2

1Penn State2Copenhagen Business School

November 8, 2015

PRELIMINARY

Inequality has long fascinated economists, and growing income inequality has been recently

and heatedly discussed in public forums.1 A remarkable fact emerging from this discussion

is the strong positive relationship between wage inequality and city size (Baum-Snow and

Pavan, 2012). In this paper, we add to the study of inequality and distribution of economic

activity in two ways. First, we document new facts on the interaction of geography with

inequality. Second, we develop a quantifiable model which explains our findings and their

implications. In particular, we study the effects of improving trade infrastructure on wage

and welfare inequality.

Our first contribution is to document facts about inequality and geography. We assign

to each American city a measure of remoteness meant to capture its distance from all other

cities. We then show that this measure correlates negatively with the skill premium, the ratio

of the mean wage of college degree to the mean wage of non-college degree workers. That is,

wage inequality is lower in remote cities. To our knowledge, our paper is the first to document

this fact.

∗We thank Sina Smid for research assistance, and participants in seminars at Cardiff University, Copen-hagen Business School, Copenhagen University, the European Trade Study Group, and Penn State for helpfulsuggestions.

1Inequality is one of the key issues in the current US presidential race.

1

The literature has developed little in the way of a unified framework that incorporates both

geography and inequality in the spatial economy. On one hand, economic geography examines

the role of trade costs in determining spatial patterns of economic activity. This literature does

not address inequality but welfare at the aggregate. A classic example is (Krugman, 1991),

and a more recent one is Allen and Arkolakis (2014). On the other hand, recent models of

spatial inequality often treat cities as isolated. Workers in a city interact with each other but

cities in a nation either do not trade or trade costlessly (Davis and Dingel, 2014). Since the

geographic location of a city relative to other cities remains irrelevant, this literature cannot

address the interaction of geography with inequality. By including costly trade between cities

in a model of mobile heterogeneous labor, we bridge the gap.

In our model, we have a continuum of locations, workers, and firms. Workers come in

two types, skilled and unskilled, and each worker has an ideosyncratic utility from living in

each location. A worker decides where to live taking wages as given. A firm also takes local

wages as given, and produces a tradable good using skilled and unskilled labor as inputs.

The sole difference between the two types of labor is that skilled labor benefits more from

agglomeration.2

We require a model which generates skill premia differing across locations. In particular,

to match our empirical findings we want less remote cities to have higher skill premia. We

develop a model with two critical features which delivers the required relationship. The

two critical features are stronger agglomeration forces for skilled workers, and heterogenous

location preferences.

The intuition behind the model can be described in a few sentences. Consider a city

near other cities, a centrally-located city. Its access to cheap tradable goods and nearby

markets make it attractive to live in. This leads the city, all else equal, to have a relatively

high population compared with a remote city. Due to agglomeration forces, skilled workers

become relatively more productive in the centrally-located city. Thus, all else equal, firms

there demand a relatively high share of skilled workers as inputs. In order for the relative

share of skilled workers on the supply side to equal the firm’s demand for relative share of

skilled workers as inputs, there must be a higher skill premium in the centrally-located city.

Our results show that improving trade infrastructure benefits both types of labor, but low

skill labor may benefit more. Better infrastructure tends to spread populations out, so that

skilled workers lose some of their agglomeration advantage over unskilled workers. Typically

2We allow there to be a Hicks-neutral agglomeration force as well.

2

models of inequality and trade predict that skilled workers gain more than unskilled workers

(Antras et al., 2006), a result that may be reversed in our model.

The literature on the skill premium has found a number of patterns. To the extent that

we are able to measure the relevant variables, all of these facts are consistent with our data.

The literature has shown convincingly that the skill premium is higher in cities with larger

populations (Davis and Dingel, 2012). The relationship between the skill premium and city

size has become stronger over time (Baum-Snow and Pavan, 2013; Lindley and Machin, 2014).

In addition, larger cities have been shown to have a higher share of college-educated worker,

another pattern which has strengthened over time (Moretti, 2008; Lindley and Machin, 2014).

Areas of denser economic activity have higher skill premiums (Combes et al., 2012).

These stylized facts have inspired a number of theories (Davis and Dingel, 2012, 2014;

Baum-Snow and Pavan, 2012; Combes et al., 2012). These theories abstract from costly trade,

treating trade between cities as either non-existant or frictionless.3 The style of our modeling

exercise below has more in common with recent forays of trade economists into economic

geography (Allen and Arkolakis, 2014; Desmet et al., 2014; Fajgelbaum et al., 2015). These

theories model costly trade, and focus on the spatial location of economic activity. They have,

however, only one type of labor, so they cannot analyze the interaction between geography

and inequality. In an international trade context, Fujita and Thisse (2006) study inequality

with costly trade, but their model has immobile unskilled workers.

A recent working paper, Fan (2015) analyzes the impact of an international trade liberal-

ization on inequality using a spatial equilibrium framework. As Fan’s focus is on aggregate

welfare effects rather than on the forces determining the distribution of the skill premium, he

relies on skilled and unskilled workers differentially valuing amenities to generate the observed

distribution of the skill premium.4 Since we are interested in the geographic distribution of

inequality, we have a simpler model in which skilled workers only differ from the unskilled in

terms of strength of agglomeration effects in the production function.

3Davis and Dingel (2012) does contain an extension where trade costs are treated in the limit as they goto zero.

4Skilled and unskilled workers in Fan (2015) differ in terms of valuation of amenities, draws from productiv-ity distributions, costs of migration, and initial allocations across space. The initial allocations are importantas migration costs depend on the location a worker starts from.

3

1 Documenting inequality and geography

In this section, we describe our data sources, give our definitions of measures of geography

and inequality, and present the empirical findings which motivate our modeling exercise.

1.1 Data sources

Our empirical section is largely based on the IPUMS 5% sample of the 2000 American census.

In this cut of the data, we use individuals older than 24 and younger than 65 with reported

income, giving us observations on over four million workers distributed across the United

States.5

We want to compare inequality in different locations. As agglomeration will be an impor-

tant component of our model, the size of a location will be critical for our analysis. Different

authors in the literature have used different regions as units of analysis. For our purposes, a

location will be a metropolitan statistical area (MSA) when possible, and otherwise a census-

related area known as a public use microdata area (PUMA). States choose PUMAs to contain

between 100,000 and 300,000 residents. PUMAs are diverse in geographical size as the popu-

lation density of American regions vary widely.

Many authors in the urban economics literature have used the same IPUM’s 5% sample,

and in the course of cleaning and understanding the data for our project we discovered some

important data issues which have received little discussion in the literature. In particular,

IPUM’s data only reliably report a PUMA for each individual. An individual’s MSA, CBSA,

and county are only reported when there is no ambiguity about her location. If an individual

is in a PUMA which straddles the border of two counties, then she will be reported without

a county. In practice, only 66% of our cleaned observations have a reported county, and

we observe individuals in only 423 of the 3007 American counties. Particularly for sparsely

populated and smaller geographic areas, the IPUM’s identifiers are likely unrepresentative.

For MSA’s, it is likely that our analysis will pick up the core of the urban area, but miss those

on the perimeter.

In addition to the IPUMs data, we need information on the geographical position of each

location as well as information on trade flows between locations. We use geographical position

data from the Missouri Census Data Center. For trade flows we use publicly-available data

5We clean the data using modified replication code from Baum-Snow and Pavan (2013). For more infor-mation on how the data were cleaned, see the data appendix.

4

from the US Commodity Flow Survey. Our data on trade flows is from 2007, as this is the first

year in which data is available at the required level of disaggregation. For a more complete

discussion of data sources and manipulation, see the data appendix.

1.2 Important concepts

Our main measure of geography is remoteness, a concept we borrow from the international

trade literature.6 We label each location with a number i = 1 . . . N . The distance between

location i and location j is di,j. The distance we use here is structurally estimated later in

this paper, and captures the trade cost between every pair of locations given the current state

of transportation infrastructure. The remoteness of location i Ri is the harmonic mean of the

distances between location i and the other locations:

Ri =1∑

j 6=i1di,j

In words, a location which is close to other locations will have low remoteness. An al-

ternative formulation a bit closer to the trade literature would be to weight the distances

by population (Head, 2003). A weighted measure is more appropriate in the trade context

where population is immobile to a first approximation. In our context, however, population

in endogenous and depends upon trade costs, so we stick to the unweighted version.7

We have two measures of inequality, the college wage premium (or skill premium) and

the college population share (or skill share). Both are measured in a way standard in the

labor literature, and are calculated in each location. A worker is highly educated if he has at

four-year college degree. The college premium is the mean wage of highly-educated workers

in an location divided by the mean wage of other workers. The college population share is

the population of highly-educated workers in an location divided by the population of less-

educated workers. We use census population weights when calculating all means.

6It is a famous empirical regularity that the trade of two countries is proportional to the product of theirnational products divided by the physical distance between them. This relationship is known as the naivegravity equation. The adjective naive makes an appearance because such a gravity equation implies that tradebetween two countries is unaffected by what takes place in a third country. For example, the trade between theUnited States and Mexico is unaffected by the rapid increase in trade between China and the United States.Remoteness in an international trade context is the national-product-weighted sum of the distance between acountry and all other countries. Multiplying the gravity equation by a remoteness term allows third countriesto influence bilateral trade relationships.

7We have done the analysis with weighted versions of this measure, and the results are similar.

5

Both in this motivating empirical section and the structural model, we will be focusing

on population density rather than population as the driver of agglomeration and congestion.

This is necessary because of the census requirement that the PUMA, which is the smallest

area we can observe, contains between 100,000 and 300,000 people. Thus the population of

every PUMA is nearly. The population density, on the other hand, varies by several orders

of magnitude in a meaningful way. For example one PUMA in Montana has 0.17 people per

square km. in a PUMA in Montana, while another in Connecticut has 311 people per square

km, similar to the New York City MSA.

Table 1 contains some descriptive statistics, and the series of figures Figure 1 shows how

our measures vary across the United States. The borders on this map are PUMAs. PUMAs

within an MSA are just all assigned the data value for the MSA. The more intense the blue,

the higher the measure. Remoteness is highest in the West, and particular the North West

of the United States. Population density is highest in metropolitan areas. The college wage

premium seems to be higher in the parts of the country which are relatively less remote, a

finding we will pick up in our regressions in the next section. The college population share,

however, is relatively high in the Rocky Mountain states such as Montana and Idaho, and not

clearly correlated with remoteness. This will also be our finding in the next section.

Statistic Mean Standard Dev Min Max

Remoteness 2.5×10−3 3.3×10−4 2.0×10−3 3.2×10−3Population Density 25.7 48.2 0.17 408Skill premium 1.58 0.14 1.20 2.26College share 0.32 0.16 0.11 1.34

Census observations 4.3×106

Location observations 775

Table 1: Data summary statistics

1.3 Motivating empirical results

In this section we document the covariance of our measure of geography, remoteness, with our

measures of inequality, the college wage premium and college share. Regressions of the skill

premium on remoteness and other variables are found in Table 2. In column one, we see that

remoteness has a strong negative relationship with population density. Doubling a locations

remoteness is associated with a fourfold decrease in its population density.

6

(a) Remoteness (b) Population Density

(c) Skill Premium (d) College Share

Figure 1: Counties by data feature

Remoteness also covaries negatively with the skill premium, with an elasticity of around

12%. To put this in context, we find that the population density of a location varies positively

with the skill premium with an elasticity of around 2%. The positive relationship between

city size and the skill premium has been emphasized in other studies. We find that both

remoteness and population density are significantly and positively associated with college

population share. The correlation of remoteness with college population share is not, however,

robust to different regression specifications. For example, in Table 3 we use simple geodesic

distance rather than our structurally estimated distance. The correlation between population

share and remoteness remains positive, but loses statistical significance.

We suspect that a few land-grant university outliers are behind the unstable positive

correlation between remoteness and college population share. The most remote locations in

7

(1) (2) (3) (4) (5)VARIABLES L pop dens L wage prem L wage prem L coll shr L coll shr

L remote -3.681*** -0.124*** 0.569***(0.450) (0.0255) (0.116)

L pop dens 0.0211*** 0.151***(0.00203) (0.0105)

Constant -19.87*** -0.292* 0.405*** 2.177*** -1.591***(2.723) (0.153) (0.00554) (0.695) (0.0280)

Observations 775 775 775 775 775R2 0.101 0.031 0.120 0.025 0.231

Robust standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1

Table 2: Regressions of log college premium on log geography variables

our sample include Iowa City, IA; Santa Fe, NM; Ames, IA; Lincoln, NE; Albuquerque, NM;

and Missoula, MT. Iowa City is the location with the highest college population share in the

United States, and the other locations on this list also contain high college population shares.

Farid’s affiliation, Penn State, is clearly visible as the college population share outlier in the

center of Pennsylvania. These relationships are not surprising because each of these locations

contains a large university or national laboratory. Remote universities are vestiges of a 19th

century US policy encouraging each state to fund its own public university, and the Sandia

and Los Alamos laboratories at Santa Fe were deliberately located in remote locations due to

secret research. Since these exogenous shocks to high skilled productivity in remote locations

are outside of our model, we do not try to capture the possible positive correlation between

remoteness and college population share in our theory.8

The takeaway from the empirical section is that geographical position seems to have a

strong effect on inequality. In particular, relatively remote locations in the United States have

relatively low population density and low college wage premia. These two facts motivate the

theory developed in the next section.

2 Theory

In the last section, we presented evidence that geography plays a role in shaping the college

wage premia. In this section we generalize spatial models of trade to explain our findings

as well as the previously documented fact that college population shares and college wage

8There is more work to be done excluding these public university outliers and reestimating.

8

(1) (2) (3) (4) (5)VARIABLES L pop dens L wage prem L wage prem L coll shr L coll shr

L remote -0.703*** -0.0115* 0.0428(0.0957) (0.00603) (0.0321)

L pop dens 0.0211*** 0.151***(0.00203) (0.0105)

Constant -3.659*** 0.356*** 0.405*** -0.887*** -1.591***(0.821) (0.0512) (0.00554) (0.270) (0.0280)

Observations 775 775 775 775 775R2 0.066 0.005 0.120 0.002 0.231

Robust standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1

Table 3: Regressions of log college premium on log geography variables, geodesic distance

premia are positively correlated with population size. In particular, the model introduces

a mechanism to explain why the college wage premium and population decreases with the

remoteness of a location.

2.1 Environment

The model is static, with a continuum of locations j ∈ J , a continuum of skilled workers with

total population NH , and a continuum of unskilled workers with total population NL. Workers

can choose to reside and work in any (single) location. Firms in each location can produce a

location-specific variety of a tradable final good with a constant elasticity production function

using the two types of labor as inputs. Both workers and firms are price takers in perfectly

competitive markets.

2.2 Worker’s problem

The utility of a worker in location i is:

Q(i)u(i)ε(i) (1)

where Q(i) is utility from tradeables, u(i) is the utility from exogenous amenities, and ε(i)

is worker-specific preference for location i. Tradeable goods are differentiated by the origin

of production, and q(j, i) is consumer’s consumption in i from goods originated in location j.

9

The aggregator Q(i) in the utility function with elasticity of substitution σ > 0 is:

Q(i) =

[∫J

q(j, i)σ−1σ dj

] σσ−1

,

We assume that worker location prefrences ε(i) are independent across workers and locations,

and have a Type II Extreme Value distribution,

Pr(ε(i) ≤ x) = exp(−x−θ)

Workers are endowed with exogenous skill s ∈ {H,L}. A worker’s wage in location i is denoted

by ws(i), and her budget constraint is

ws(i) =

∫J

p(j, i)q(j, i) dj (2)

where p(j, i) is price of good j in destination i, and J is the given set of locations. The only

place skill type appears in the worker’s problem, is through the wage’s effect on the budget

constraint. Otherwise workers of both types have the same preferences.

A worker has two types of decision to make. She decides where to live, and how much of

each good to consume. Given a choice of location, the second problem is standard. A worker

of type s in location i spends xs(j, i) on goods produced in j,

xs(j, i) ≡ p(j, i)qs(j, i) =[p(j, i)P (i)

]1−σws (3)

where P (i) is the CES price index,

P (i) =

[∫J

p(j, i)1−σ dj

] 11−σ

(4)

The second decision a worker makes is where to live. In order to characterize this decision,

we introduce congestion forces as in Allen and Arkolakis (2014):

u(i) = u(i)n(i)γ (5)

Here, n(i) is total population in location i and γ < 0 is the degree of congestion effect. This

10

specification is isomorphic to a setting where preferences are Cobb-Douglas between tradeables

and housing in which housing supply is inelastic. A worker with skill level s faces the following

discrete choice problem over locations:

maxi∈J

ws(i)

P (i)u(i)ε(i)

Using the properties of the Type-II extreme value distribution, we can characterize the share

of type-s labor in location i as:9

ns(i)

ns(j)=

(ws(i)u(i)/P (i)

ws(j)u(j)/P (j)

)θ(6)

where ns(i) is population of workers with type s in location i. The elasticity of relative labor

supply to relative wages is:

∂(ns(i)/ns(j)

)/(ns(i)/ns(j)

)∂(ws(i)/ws(j)

)/(ws(i)/ws(j)

) = θ

The variance of ε(i) across both workers and locations is decreasing in θ. A large θ implies

that individual unobserved preferences for location are similar across locations. Small changes

in wages, prices, or amenities induce large movements of workers. Another way of putting it

is that the supply curve of workers to a location is flat. When θ is small, workers have widely

varying preferences over locations, so that large changes in wages, prices, or amenities are

necessary to induce movement. That is, the supply curve of workers to a location is steep.

A location with a higher relative wage for skilled workers will attract relatively more

skilled workers, but it won’t attract all the skilled workers. Small changes in relative wages

will induce small changes in relative populations. In a model without preference heterogeneity,

9Incidentally this equation gives the model its first testable prediction:

wH(i)wL(i)

nH(i)nL(i)

1θ

=

wH(j)wL(j)

nH(j)nL(j)

1θ

A higher ratio of skilled workers to unskilled workers implies a higher skill premium in wages. In the data,the college wage premium and college population share are strongly positively correlated. A simple regressionimplies a θ value of 1.5.

11

in equilibrium all populated locations must offer the same utility within skill type. This means

that a location which deviates and offers a slightly higher wage will attract all workers.

Define the well-being index, denoted by Ws, for population of skill s:

Ws ≡

[∫ ‖J‖j=0

(ws(j)P (j)−1u(j)

)θdj

] 1θ

This index is proportional to the expected welfare of a worker of type s before she draws her

location preferences.10 Again using properties of the Type-II extreme value distribution, we

derive the relationship for all i ∈ J :

ns(i)

Ns

=

(ws(i)P (i)−1u(i)

Ws

)θ(7)

Here Ns is the exogenously given total population of type-s workers. If a location offers rela-

tively high welfare, it will have a relatively high population, with the extent of the relationship

governed by θ.

2.3 Firm’s problem

Each location has a representative firm with a CES production function using skilled and

unskilled workers,

A(i)[βH(i)nH(i)

ε−1ε + βL(i)nL(i)

ε−1ε

] εε−1

where A(i) is total factor productivity in location i. ε > 0 is the elasticity of substitution

between high and low skill workers. βH(i) > 0 and βL(i) > 0 are factor intensities. Differences

in factor productivities (and agglomeration, to be discussed shortly) are the only differences

between skilled and unskilled labor. Following the literature, we distinguish two agglomeration

forces. First

A(i) = A(i)n(i)α (8)

10To get the actual welfare, we must multiply by the gamma function Γ(1 + 1θ ). This scaling term depends

only upon θ, an exogenous preference parameter.

12

with α > 0. This agglomeration force changes productivity of both low and high skill work-

ers. A standard Krugman type monopolistic competition with free entry generates the same

relation through endogenous measure of firms (if α = 1/(1 + σ)). In addition, there is strong

evidence that agglomeration forces are stronger for high skill workers (Glaeser and Resseger,

2010). Motivated by these findings, we make a skilled worker’s productivity covary positively

with the population of skilled workers in a city:11

βH(i) = βH(i)nH(i)ϕ

βL(i) = βL(i) (9)

Here, ϕ > 0 governs the skilled bias technological change for high skilled labor. By cost

minimization, the unit cost of production equals

c(i)

A(i), where c(i) =

[βH(i)εwH(i)1−ε + βL(i)εwL(i)1−ε

] 11−ε

(10)

The share of spending of producers on high skill workers, denoted by b(i), is given by

b(i) =βH(i)εwH(i)1−ε

βH(i)εwH(i)1−ε + βL(i)εwL(i)1−ε(11)

As βH(i) is increasing in the population of skilled workers in a city, so is b(i).

Now the intuition for our result is available. Suppose a new highway is built to a city.

As a result of reductions in trade costs, the city will benefit from a lower price index and

higher sales abroad, so firms in the city will attract workers of both types. As the city

grows, productivity rises. However, due to agglomoreation advantages, skilled workers become

relatively more productive. Firms will demand relatively more skilled workers. There will be

a disequilibrium as firms demand relatively more skilled workers than the skill share in the

population. Equilibrium is restored by raising skilled wages, which will both attract skilled

workers and discourage firms from hiring them.

As mentioned above, markets are perfectly competitive. Let d(j, i) be the trade costs of

shipping a good from j to i. The price of a good produced in location j and consumed in

location i is:

11For both productivities and amenities, we have been talking about population while in our empiricalexercise we used population density. From (8) and (9), it can be seen that the area is just a constant whichwill be absorbed by the levels and could in principle be easily adjusted for.

13

p(j, i) = c(j)d(j, i)/A(j) (12)

2.4 Spatial Equilibrium

A spatial equilibrium is a set of wH(i), wL(i), nH(i), and nL(i) such that:12

1. Labor shares in locations are according to (7).

2. Demand and supply of skilled and unskilled labor are equal in each location.

3. Goods market clear

wH(i)nH(i) = b(i)

∫J

∑s∈{H,L}

ns(i)xs(j, i) dj (13)

where xs(j, i) is given by (3), and b(i) is given by (11).

4. Allocation of labor is feasible,∫JnH(j) dj = NH and

∫JnL(j) dj = NL. Wages are

normalized∫JwH(j)dj = 1.

2.5 Equilibrium Analysis

Before we derive a system of equations characterizing the equilibrium, we first discuss some

of its features. In particular, we show that without both heterogeneous location preferences

and skill-specific agglomeration, our model would not generate location-specific skill premia.

However, we need no other type of skill-specific preferences, productivity, or mobility to

generate this result.

Perfect competition gives us that the income share of skilled labor equals firm’s optimal

input share on skilled labor:

wH(i)nH(i)

wH(i)nH(i) + wL(i)nL(i)=

βH(i)εwH(i)1−ε

βH(i)εwH(i)1−ε + βL(i)εwL(i)1−ε

12There is a bit hidden in this definition, as we have already used perfect competition to give us goods pricesp(j, i) given wages, the consumer’s optimization problem to give us demand x(j, i) given prices and wages,and the firm’s optimization problem to give us relative labor demand (11)

14

With some algebra, and substituting from (9), we get:

nH(i)

nL(i)=

(βH(i)

βL(i)

)ε(wH(i)

wL(i)

)−εnH(i)ϕε (14)

Using equation (7), the supply side of labor markets imply:13

nH(i)

nL(i)=NH

NL

(WH

WL

)−θ (wH(i)

wL(i)

)θ(15)

Labor markets clear when skill premia simultaneously satisfy the pairs of demand (14) and

supply (15). Combining, we get:

wH(i)

wL(i)=

(WH

WL

) θθ+ε(NH

NL

) −1θ+ε(βH(i)

βL(i)

) εθ+ε

nH(i)ϕεθ+ε (16)

Suppose we were to shut down preference heterogeneity, θ → ∞. From (16) we see that the

skill premium will be constant across locations (and, surprise, proportional to ex-ante expected

welfare). On the other hand, suppose there is no agglomeration for skilled workers, ϕ = 0.

Then the skill premium can vary between destinations only due to exogenous differences in

returns to skilled and unskilled labor. In order to have equilibria with (endogenously) varying

skill premia, we need both θ to be finite and ϕ > 0. In words, we need both variance in

unobserved location preferences and the agglomeration force for skilled workers. Large cities

demand more skilled workers due to agglomeration, but when unobserved location preferences

matter skilled workers do not fully arbitrage the a wage increase away. In this respect, our

model departs from a standard Rosen-Roback (or Allen-Arkolakis) model by highlighting the

interaction of agglomeration and the labor supply elasticity.

Next we show that the distribution of workers across space has a real effect on welfare

inequality. Plugging equilibrium skill premium from (16) into relative labor supply (15),

distribution of low skilled labor can be written as a function of distribution of high skilled

labor,

nL(i) =( βH(i)

βL(i)

)−θεθ+ε(WH

WL

) θεθ+ε(NH

NL

) −εθ+ε(nH(i)

) θ(1−εϕ)+εθ+ε

(17)

13As far as we are aware, other models of inequality in the literature assume fundamental differences inpreferences between skilled and unskilled workers. To derive (15) we are using the fact that skilled andunskilled workers value amenities in the same way, and that they face the same prices.

15

Define ρH(i) = nH(i)/NH as the density of high skill labor. Equation (17) andNL =∫JnL(i) di

pin down the relative average well-being of skilled and unskilled workers, WH/WL, as a func-

tion of ρH(i),

WH

WL

=(NH

NL

)− 1ε︸ ︷︷ ︸

aggergate scarcity

× (NH)ϕ︸ ︷︷ ︸aggregate agglom.

×

[∫J

( βH(i)

βL(i)

)−θεθ+ε(ρH(i)

) θ(1−ϕ)+εθ+ε

di

]− θ+εθε

︸ ︷︷ ︸distributional effect

(18)

Equation (18) decomposes the three forces behind real well-being inequality: 1) Aggregate

scarcity of high skill labor, 2) Aggregate agglomeration advantage of high skill workers, 3) A

weighted average of relative exogenous productivities in which weights are determined by the

the distribution of the density of high skill labor. While the first two behave at the aggregate,

the third relates to the distribution of skills. In this sense, Equation (18) relates an index

of real inequality at the national level to the distribution of high skill workers across cities

within the nation.

2.6 Equilibrium equations

In this section we characterize our equilibrium with only two integral equations. These integral

equations will be the basis of our empirical exercise.

By simply inverting the definition (11) of input share b(i), we get total income in i equal

to 1b(i)wH(i)nH(i). Using the goods market clearing condition, equation (13), we derive:

wH(i)nH(i)b(i)−1 =

∫J

[c(i)d(i, j)

A(i)P (j)

]1−σwH(j)nH(j)b(j)−1 dj

Equations (10) and (11) further imply that

c(i) = c(i)wH(i)

where c(i) =[βH(i)nH(i)ϕ

] ε1−ε[b(i)] −1

1−ε(19)

Replacing c(i) with c(i) in the above integral equation and also replacing the price index using

16

(7), after some algebra we get:

A(i)1−σ c(i)σ−1nH(i)wH(i)σb(i)−1

= W 1−σH N

σ−1θ

H

∫J

d(i, j)1−σu(j)σ−1nH(j)(θ+1−σ)/θwH(j)σb(j)−1 dj (20)

Again substituting the price index from (7) and c(i) into (4), we get another integral equation:

u(i)1−σnH(i)(σ−1)/θwH(i)1−σ =

W 1−σH N

σ−1θ

H

∫J

d(j, i)1−σA(j)σ−1c(j)1−σwH(j)1−σ dj (21)

Equations 25–26 give us two systems of integral equations. If ϕ = 0 and θ =∞, the integral

equations collapse to those in Allen and Arkolakis (2014).

We can reduce our system further using a method from Allen and Arkolakis (2014). If

either of the systems of integral equation hold along with the following relation, both systems

of integral equations must hold:

b(i)−1A(i)1−σ c(i)σ−1nH(i)wH(i)σ = λu(i)1−σnH(i)(σ−1)/θwH(i)1−σ (22)

where λ > 0 is some constant. The solution algorithm developed below is based on successive

iterations over nH(i).

There are two “endogenous” variables hidden in (25) - (22), b(i) and c. Endogenous here

means that they depend upon nH(i). Thus solving our model involves an inner loop to update

these endogenous variables. Details of our solution algorithm are contained in Appendix C.

Allen and Arkolakis (2014) have a slick proof of uniqueness, which depends only on the

relationship between parameters governing agglomeration and parameters governing conges-

tion. Unfortunately the same proof cannot be used in our setting. Intuitively, there should be

a similar condition in our model relating the agglomeration forces to the congestion force will

be that the parameter governing joint agglomeration α and the parameter governing skilled

agglomeration ϕ cannot be too large relative to the parameter governing congestion γ.

17

3 Estimation

In this section we discuss the estimation of our structural model. Trade costs are estimated

first, followed by a loop which estimates the base productivities and amenities of each location

given observed wage premia and college population shares.

3.1 Estimation of trade costs

In many countries, the largest cities are on coastlines or near major rivers. The United States

is no exception, with the East and West coasts containing the majority of the population. If

one thinks exogenous trade costs are simply quadratic in distance, then one might think a

location in the center of the United States would have the lowest trade cost. The geographical

center of the United States is just outside of Lebanon, Kansas, population 218.

The flaw with this line of thinking is that the geography affects the cost of trading between

any two locations. It is often easier to go around a mountain even if the geodesic between two

locations goes through one. New York and Miami are about as far apart as New York and

Lebanon, Kansas, but it is natural to think that shipping bagels to Miami is cheaper because

of the possibility of using a ship. To this end, we estimate trade costs by using a method from

Allen and Arkolakis (2014) which takes geographic features into account. We provide a short

overview here, with more detail in the data appendix and in the origin Allen and Arkolakis

paper.

There are three steps to the estimation process. In the first step, we use three separate

image files each showing a map of the United States. On one of the maps is the road network,

on the second is the railway network, and on the last is the waterway network. We consider

four possible methods for moving goods – road, rail, water, and air. For each of these methods

separately, we assign a cost of traveling over each pixel of the relevant image file. For example,

if we are considering water transport, we assign a low cost to each water pixel, and a high

cost to all other pixels. Then, we calculate the lowest possible cost of using each method to

move goods between all pairs of locations. The algorithm we use to find this lowest cost path

for each transport method is called the fast marching algorithm.

After we finish the first step, we know how much it costs to move goods on the road between

two locations, but only in terms of the units we assigned to road travel. We cannot compare

the cost of road travel to the cost of water transport because we don’t know the exchange

rate, as it were, of road travel to water transport. The second step is to use a discrete

18

choice framework and the fraction of observed trade flows via each mode between each pair

of locations in order to back out these exchange rates. The idea is that if a large share of

transport is via road, then it must be that road is a relatively cheap form of transportation.

At the end of the second step, we have recovered the exchange rates, or if you like relative

costs, between each of the forms of transport. It is not enough for our structural model,

however, to know that it costs twice as much to move goods by road as it does to move them

by air. We need to pin down the level of costs as well. To do this we return to a trade classic,

the gravity model. The version we use is given by combining (3) and (12) to get exports from

location j to location i:

X(j, i) =[c(j)d(j, i)

A(j)P (i)

]1−σWi (23)

Here Wi is the total income of workers in location i. Take logs, and the only bilateral term

in (23) is the distance cost. Using the discrete choice model, we can get the average cost of

transport up to an unknown parameter. Running a regression of the logged version of (23)

with origin and destination fixed effects and using the average cost of transport for d(j, i) pins

down the scale of trade costs.

3.2 Estimation of other parameters

This section is concerned with estimating and calibrating all universal and location specific

parameters, which are listed in Table 4). First we discuss the parameters we calibrate from the

literature, then parameters we get from reduced form estimates, and then finally parameters

we estimate from the equilibrium integral equations of our structural model.

3.2.1 Parameters taken from the literature

Two critical parameters for our model are the consumer’s elasticity of substitution across

goods σ and the non-skill specific agglomeration force α. Allen and Arkolakis (2014) set

σ = 9, but Monte et al. (2015) set σ = 4. Allen and Arkolakis justify their relatively high

choice with the intuition that goods produced within a country are more substitutable than

than goods produced in different countries. The Allen and Arkolakis model is isomorphic to

Monte et. al if α = 1/(σ − 1) (see appendix 2 of Monte et. al). In Monte et. al, then, the

implied α = 1/(4− 1) = 0.33, while Allen and Arkolakis choose α = 0.1 citing Rosenthal and

Strange (2004) who report that productivity rises by 3-8% when population doubles.

19

universal parameters:σ elasticity of substitution across goodsε elasticity of substitution across low- and high-skilled laborγ congestion strengthα common agglomeration strengthθ dispersion parameter of unobserved preferences for locationsϕ high-skill agglomeration advantage

location-specific parameters:βH(i) basic high-skill factor intensity in location i, βL(i) = 1− βH(i)A(i) basic TFP in location iu(i) basic amenity in location i

Table 4: List and decription of all model parameters

Gottlieb and Glaeser (2009) classify agglomeration forces in the literature with three cat-

egories: (1) urban concentration brings about a reduction in transport costs; (2) urban con-

centration facilitates labor movement across firms; (3) urban concentration encourages the

exchange of ideas and innovation. The first two forces apply to both types of labor, while

the third is more important for high skilled labor. Recent studies of the wage premium have

emphasized the exchange of ideas (Davis and Dingel, 2012; Baum-Snow and Pavan, 2013).

Since σ and α will affect welfare, we choose a relatively high σ = 9 and a low α = 0 within

the ranges suggested by the literature because this will tend to bias our welfare estimates

down.

Value of γ. In Mante et al., preferences are Cobb-Douglas over differentiated goods and

housing, with the share of housing be equal δ. Again, an isomorphism with Allen and Arkolakis

requires that congestion γ = δ/(1 − δ). Allen and Arkolakis report that, according to the

Bureau of Labor Statistics (BLS), the housing expenditure γ in the year 2000 was in the range

of 19%–25% depending on which items you count as housing. This implies that congestion γ in

the range of -0.23 to -0.33. Allen and Arkolakis choose γ = −0.30. On the other hand, Monte

et al. report that δ = 0.40 according to the Bureau of Economic Analysis, so γ = −0.67. In

Diamond (2015), housing share in expenditures is in the range of 0.39 to 0.46 according to the

year 2000 Consumer Expenditure Survey. In Moretti (2013) housing share is 42.7% according

to the BLS (counting fuel and utilities as housing). The bottom line is that δ ∈ [0.35, 0.40] is

in line with the literature, so we choose γ = −0.6.

Value of ε. Katz et al. (1999) provide a literature review where they report values for the

elasticity between high and low skill labor in the production function of 1.40 to 1.70. Ciccone

20

and Peri (2006) come up with estimated values of elasticity of substitution between unskilled

and skilled labor between 1.3 and 2. Diamond (2015) estimates ε = 1.6. Card (2009) using

data on MSA’s finds that ε = 2.5. There are many other similar estimates.

One notable exception is Baum-Snow et al. (2014). They use CBSAs, and their elasticity

ε is the elasticity between capital-augmented high-skilled labor and low-skilled labor. In their

reduced form estimation they find ε roughly in the range of 1.5 to 2.5. But in their structural

estimation, they find ε ∈ [6.7, 10]. To be consistent with a large amount of the literature,

however, we choose ε = 2 which lies smack in the middle of the range of 1 to 3.

3.2.2 Parameters pinned down by reduced-form estimations

We can directly read a number of parameters through reduced form estimates.

Value of θ. Let ω(i) ≡ wH(i)/wL(i) be the skill premium in location i. The model implies,

log(ω(i)

ω(j)

)=

1

θlog( nH(i)/nL(i)

nH(j)/nL(j)

),

which delivers the following stochastic form:

logω(i) = α0 + α1 log(nH(i)/nL(i)

)+ errorij (24)

where in the model, α0 = logω(1) − α1 log(nH(1)/nL(1)

), α1 = 1/θ; with j = 1 as the

reference, since we have only J − 1 independent observations.

Value of ϕ and βH . At the equilibrium of labor market (equation 15),

wH(i)

wL(i)=

(WH

WL

) θθ+ε(NH

NL

) −1θ+ε(

βH(i)

1− βH(i)

) εθ+ε

nH(i)ϕεθ+ε ,

which delivers the exact following regression:

logω(i) = α0 + α1 log nH(i) + ζ(i).

21

Given ε and θ, which are already known by previous sections,

ϕ =α1(θ + ε)

εβH(i)

1− βH(i)= exp

[ζ(i)(θ + ε)

ε

]3.2.3 Parameters pinned down by the system of equilibrium equations

Finally, the inherent, base values of productivity and amenities will be estimated from theequilibrium integral equations implied by our model. We rewrite the two systems of integralequations:

A(i)1−σ = W 1−σH N

σ−1θ

H c(i)1−σnH(i)−1wH(i)−σb(i)n(i)(σ−1)α

∫J

d(i, j)1−σu(j)σ−1n(j)(σ−1)γnH(j)(θ+1−σ)/θwH(j)σb(j)−1dj (25)

u(i)1−σ = W 1−σH N

σ−1θ

H nH(i)(1−σ)/θwH(i)σ−1n(i)(σ−1)γ

∫J

d(j, i)1−σA(j)σ−1n(j)(σ−1)αc(j)1−σwH(j)1−σdj (26)

Here, A(i) and u(i) are the only unknowns, while all parameters are already estimated/calibrated

and all other variables are given by data. We can further reduce the two systems of equation

into one. As in Allen and Arkolakis (2014), it is necessary that trade costs are symmetric

d(i, j) = d(j, i). Consider the following equation,

u(i)σ−1n(i)(σ−1)γnH(i)(θ+1−σ)/θwH(i)σb(i)−1 = λA(i)σ−1n(i)(σ−1)αc(i)1−σwH(i)1−σ (27)

Along with (27), system (25) of integral equations delivers (26). λ > 0 is a constant. The

numerical algorithm by which we solve these equations is described in Appendix B.

4 Results

The trade cost estimates we get are reported in Table 5, and our other calibrated and estimated

parameters are given in Table 6. Our value for preference dispersion θ is lower than that

found in Allen and Arkolakis (2014), two versus ten. This means that our individuals will

have stronger opinions about location, and wage differentials will have to be higher to induce

movement. Our trade costs are also qualitatively a bit different, especially water has a higher

variable cost in our estimation.

22

0.44 Road variable cost0.31 Rail fixed cost0.40 Rail variable cost0.34 Water fixed cost0.68 Water variable cost0.49 Air fixed cost0.32 Air variable cost

Table 5: Trade cost estimates

universal parameters:σ 9 elasticity of substitution across goodsε 2 elasticity of substitution across low- and high-skilled laborγ -0.6 congestion strengthα 0 common agglomeration strengthθ 2.62 dispersion parameter of unobserved preferences for locationsϕ 0.085 high-skill agglomeration advantage

location-specific parameter means:βH(i) 0.45 basic high-skill factor intensity in location iA(i) 2.52 basic TFP in location iu(i) 1,309 basic amenity in location i

Table 6: Other estimated model parameters

Figure 2 contains a visualization of our estimated base amenities and productivities. The

more intense the blue, the higher the measure is. Productivities are relatively high in the

Western part of the United States, as the model needs to induce people to live in those regions

with relatively high trade costs. Amenities follows a similar pattern, with high population

density areas estimated to have high amenities. This result is also found in ?, as the model

needs to justify why cities are located exactly where they are located rather than nearby. It

is reassuring that the model chooses to attract people to the Northern and Rocky Mountain

regions with productivity rather than amenities.

In our baseline estimation, the ratio of expected high skill welfare to low skill welfare

is 1.6907. The welfare is expected because an individual’s realized welfare depends on the

particular location preference draw he receives. Suppose that the government were to start a

campaign to build highways to relatively remote parts of the United States. How would such

a policy affect the distribution of wage inequality and overall welfare inequality?

In order to answer this question, we reestimate our model after reducing the highest 30%

of trade costs, trade costs of at least two. In Figure 3, we see that welfare inequality falls with

23

(a) Amenities (b) Productivities

Figure 2: Locations by estimated value

the reduction in trade costs. In Figure 4 the largest cities have decreases in population, and

the smallest, more remote cities have increases in population.

In Figure 5, we see the geography of changes in population and wage premium after a

10% reduction in the highest trade costs. The map for change in college share looks identical.

The maps are the same, because the forces which increase population also increase the wage

premium and college share.

In many trade models, reductions in trade barriers leads to relative losses for low skill

labor. In our model, it is possible that a decrease in barriers gives low skill workers relative

gains. If we reduce the trade cost to the smallest, most remote cities, those cities become

more attractive to live in. On the margin, people move to those cities. Population inequality

decreases, which causes a loss of gains from agglomeration accruing to high skilled workers,

so low skill people gain relatively more from a reduction in trade costs.

5 Conclusion

We find that remote cities are less dense and have a relatively low college wage premium. These

stylized facts motivate the development of our spatial equilibrium model. We show that both

heterogenous location preferences among workers and stronger agglomeration forces for skilled

workers deliver the moments we find in the data. While our model delivers simple intuition,

it is rich enough to estimate using American data. After estimating our model, we find that

reducing the costs of trade for the most remote cities reduces inequality in city size, and leads

24

Figure 3: Relative welfare change reducing high trade costs

25

Figure 4: Population change after 10% reduction in high trade costs

26

(a) Population (b) Wage premium

Figure 5: Locations by change after trade cost reduction

to a reduction in the overall welfare inequality.

27

References

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29

VARIABLE NAME DescriptionYEAR Census year DATANUM & Data set numberSERIAL Household serial numberHHWT Household weightSTATEFIP State (FIPS code)COUNTY CountyMETRO Metropolitan statusMETAREA (general) Metropolitan area [general version]METAREAD (detailed) Metropolitan area [detailed version]PUMA Public Use Microdata AreaGQ Group quarters statusPERNUM Person number in sample unitPERWT Person weightSEX SexAGE AgeRACE (general) Race [general version]RACED (detailed) Race [detailed version]EDUC (general) Educational attainment [general version]EDUCD (detailed) Educational attainment [detailed version]WKSWORK1 Weeks worked last yearUHRSWORK Usual hours worked per weekINCWAGE Wage and salary incomeINCBUS00 Business and farm income, 2000.

Table 7: Variables from IPUMS 5% sample

A Data appendix

In this appendix, we describe exactly what data we used, where we got it, and how we

processed it. The goal is that a researcher wishing to replicate our analysis will be able to use

this section and code available on our website to exactly replicate and understand our results.

A.1 Cleaning and adding geography to census data

As mentioned in the body of the paper, the main data source is the IPUMS 5% sample. The

data was downloaded with the interface available on the IPUMS website.14 Table ?? described

the variables we downloaded in the initial sample.

Using these variables, we cleaned the data by modifying replication code for Baum-Snow

and Pavan (2013) from Nathanial Baum-Snow’s website.15 Cleaning involves dropping obser-

vations with imputed characteristics, dropping ages less than 25 and greater than 64, dropping

those who worked less than 40 weeks in the year, those which made less than minimum wage

14https:usa.ipums.orgusa-actionvariablesgroup15http://www.econ.brown.edu/fac/nathaniel baum-snow/ineq-citysize.zip

30

in 1999, and active duty military. Finally anyone with positive business income was dropped.

This last restriction is because we think wages are a poor measure of income for owners of

businesses. The cleaning was done in Stata with the file “census prep.do”, and cleaned data

is saved as “census00.dta”.

After cleaning the data, the next step is to merge in the latitude, longitude, and physical ar-

eas of PUMAs and MSA’s. This operation is done in Stata using the file “msa puma geography.do”.

We calculate the population weighted locations of PUMA’s and MSA’s using the excellent Mis-

souri Census Data Center website.16. Selecting all states and both “source” and “target” equal

to “PUMA for 5 Pct Samples (2000)” generates a csv file, which we copy into an excel sheet

to get “PUMAs.xlsx”. Selecting all states and both “source” and “target” equal to “Metro

Area:MSA or CMSA (2000)” generates a csv file. This csv gives MSAs four digit codes to

match CMSA codes, while in the census data we have only three digit codes. For the most

deleting the last digit of the four digit codes is all that is necessary, but in five instances

there are two four digit codes with the first three digits identical. We select the region which

matches the census MSA three digit code. These five changes are:

1. 233: 2330/2335 exclude both

2. 265: 2650/2655 exclude 2655 (Florence,SC)

3. 298: 2980/2985 exclude 2985 (Grand Forks, ND-MN)

4. 328: 3280/3285 exclude 3285 (Hattiesburg, MS)

5. 360: 3600/3605 exclude 3600 (Jacksonville FL)

After these changes, results are stored in the file “MSAs-change.xlsx”

Next we add the areas of PUMAs to complete the geographical features we need. We down-

load an IPUMS file containing all intersections between 2000 PUMAs and 2010 PUMAs.17

We then collapse this file by 2000 PUMA to get areas in square kilometers.

Finally, the file “census prep.do” merges all the geographical features into the census data,

and creates a new variable “fj region” which is an MSA if the census observation is classified in

an MSA, and PUMA otherwise. It is important to point out that just because an observation

is not classified in an MSA, that it is not in fact part of an MSA. Moreover, the population

16http://mcdc2.missouri.edu/websas/geocorr2k.html17https://usa.ipums.org/usa/resources/volii/puma00 puma10 spatial crosswalk.xlsx

31

observed in an MSA may not be representative of the true MSA population. Here is what is

said about the issue on the IPUMS website:18

The most detailed geographic information available is for 1980 county groups

or for 1990 or 2000 PUMAs, areas which occasionally straddle official metro area

boundaries. If any portion of a straddling area’s population resided outside a single

metro area, the METAREA variable uses a conservative assignment strategy and

identifies no metro area for all residents of the straddling area.

Users should not assume that the identified portion of a partly identified metro

area is a representative sample of the entire metro area. In fact, because the

unidentified population is located in areas that straddle the metro area boundaries,

the identified population will often skew toward core populations and omit out-

lying communities. Also, weighted population counts for incompletely identified

metro areas will be low by amounts ranging from 1 to 69% (since the unidentified

individuals will not be counted as living in the metro area).

A.2 Constructing CFS area distances

In order to calculate distances between Commodity Flow Survey (CFS) areas, we need two

types of information. One is information about the size and nature of commodity flows

themselves, and the second is the physical locations of roads, waterways, and railways in

the United States. Information on commodity flows was downloaded from the US Census

website.19 We used flows from the year 2007, because this was the first year in which tables

breaking down commodity flows by mode of transportation was available. These raw data

come in so-called ”long” format, with each row a origin-destination-mode observation. We

find it more convenient to work with data in the ”wide” format, with each row an origin and

destination, but with separate columns for the value of each mode of transportation. We do

this using the python script “pivot cfs mode.py”.

There are two input files necessary to run “pivot cfs mode.py”. The first, “Origin by Destination by Mode python input.csv“,

is simply the downloaded census file saved as a csv. We also need the centroid of each CFS

area in order to calculate physical distance between CFS areas. We use QGIS software to do

this. Our data comes from a shapefile available from a US census website.20 We manipulate

18https://usa.ipums.org/usa-action/variables/METAREA#description section19http://www2.census.gov/econ2007/CF/sector00/special tabs/Origin by Destination by Mode.zip20http://www.census.gov/econ/census/shapefiles/CFS AREA shapefile 010215.zip

32

VARIABLE NAME DescriptionYEAR Census yearDATANUM Data set numberSERIAL Household serial numberHHWT Household weightGQ Group quarters statusPERNUM Person number in sample unitPERWT Person weightRACE (general) Race [general version]RACED (detailed) Race [detailed version]BPL (general) Birthplace [general version]BPLD (detailed) Birthplace [detailed version]LANGUAGE (general) Language spoken [general version]LANGUAGED (detailed) Language spoken [detailed version]RACESING (general) Race: Single race identification [general version]RACESINGD (detailed) Race: Single race identification [detailed version]

Table 8: Demographic variables from IPUMS 5% sample

the data in the QGIS project ”calc centroids.qgs“. Specifically, after loading the downloaded

shapefile, we create a new variable ”st cfs area“ to make CFS areas unique. We then use

the ”dissolve“ command to eliminate counties, and the ”mean coordinates“ command to get

centroids. We save the calculated centroids as ”cfs 2007 centroids.csv“.

The output of “pivot cfs mode.py” is “replication data’ no ethnic.csv”. As the name im-

plies, in order to run the gravity equation in our distance cost estimation we need to add

information on the correlation in ethnic composition between all CFS areas. We separately

downloaded the variables in Table 8 from IPUMS. We save these new variables as the stata

file ”demographic data 2000.dta”.

The Stata script ”merge in demographics.do” combines the new and the old census data,

and then creates the correlation matrices ”lang corr matrix.csv”, ”race corr matrix.csv”, ”birth pl corr matrix.csv”.

We want the correlation matrices listed by origin destination pair to match our CFS data, so

we reshape and combine the data in this format in the python script ”append ethnic var.py”,

which creates a file ”combined stacked.dta”. Finally, we add the ethnic variables to our

CFS mode information using the do file ”append stacked.do”, which outputs the file ”repli-

cation data.csv”.

The file ”replication data.csv” now contains everything we need to run the distance cost

estimation except the map files. The last step, however, is to put the data in a format which

Matlab can understand.21 This means that we strip off all text and put the CFS area coor-

21Allen and Arkolakis wrote their estimation in Matlab and kindly made the files available to us. We use a

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dinate columns into a file called ”cfs coor.csv”, the trade value columns into ”cfs trade.csv”,

and the demographic correlations into ”cfs eth.csv”.

The Matlab script “allen arkolakis estimation.m” takes the csv files described in the last

paragraph as inputs, and outputs the file “cfs areas trade cost list.csv”. We next copy this

list as a column into the file “replication data.csv”. This is necessary because we need origin

and destination names attached to the trade costs to merge with the census data.

At this point we have recovered distance costs between all CFS areas. For the structural

estimation, however, we need to know the distance costs between all fj regions. Typically,

fj regions are completely included in a single CFS area. A small number of fj regions stradle

two or more CFS areas. In these cases, we prefer first non “rest of state” CFS areas, and then

the CFS area which contains the highest population.22 This calculation is done in the Stata

script “link.do”. The script goes on to recreate the distance cost matrix, but with fj regions

rather than CFS areas. If the distance cost is missing (i.e. there is no trade between the

relevant CFS areas), then the highest observed trade cost is substituted. The final output

is a distance cost matrix with both rows and columns fj regions. This is one input into our

structural estimation.

B Algorithm for solving integral equations

Up to scale such that∫JA(i)σ−1di = A.

1. Guess A(i).

2. Define f(i) = λ−1/2A(i)1−σ, κ = W 1−σH N

σ−1θ

H , and

K(j, i) = c(i)1−σnH(i)−1wH(i)−σb(i)n(i)(σ−1)αd(j, i)1−σn(j)(σ−1)αc(j)1−σwH(j)1−σ

Then, the system of integral equations described by (25) can be written as follows:

f(i) = κ

∫J

K(j, i)f(j)−1 dj.

modified version of their code to calculate our distance costs.22In most states there are separate CFS areas for large cities and then a single larger CFS area encompassing

the rest of the state

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In iteration t, update f (t)(i) according to

f (t+1)(i) =

∫JK(j, i)f (t)(j)−1 dj∫

J

∫JK(j, i)f (t)(j)−1 djdi

(28)

Note that as we divide integrals in 28, we do not need to know κ to update our guess.

If at some iteration t, f (t)(i) is close enough to f (t−1)(i) for all i, stop updating.

3. Find the scale parameters λ.

1 =

∫J

f(j) = λ−1/2∫J

A(j)1−σdj = λ−1/2A ⇒ λ = A1/2 (29)

While by construction∫Jf(j)dj = 1, the scale of productivity is set by A,

A(i)σ−1 = f(i)A

4. Find κ. For each i, the following is a constant equal to κ

f(i)∫JK(j, i)f(j)−1 dj

= κ

So,

WH = κ1

1−σN1θH (30)

5. Using equation (27), calculate u(i).

C Counterfactual Simulation Algorithm

1. Guess nH(i) for all i.

2. Compute WH/WL according to (18). Then plug it in (17) to find nL(i).

3. Calculate skill premia, ω(i) ≡ wH(i)/wL(i), according to (16).

4. Compute b(i) = 1/(1 + nL(i)nH(i)

1ω(i)

)

35

5. Find c(i) according to (19).

6. Calculate wH(i) according to (22) up to scale parameter λ,

wH(i) ≡ λ1

2σ−1 wH(i)

where

wH(i) = b(i)1

2σ−1A(i)σ−12σ−1 c(i)

1−σ2σ−1u(i)

1−σ2σ−1nH(i)

σ−1−θθ(2σ−1)

7. Let f(i) = wH(i)1−σ, κ = W 1−σH N

σ−1θ

H , and

K(j, i) = nH(i)(1−σ)/θu(i)σ−1d(j, i)1−σA(j)σ−1c(j)1−σ

Then, system of integral equations (26) can be written as follows (note that the scale

parameter cancels out):

f(i) = κ

∫J

K(j, i)f(j) dj

In iteration t, update f (t)(i) according to

f (t+1)(i) =

∫JK(j, i)f (t)(j) dj∫

J

∫JK(j, i)f (t)(j) djdi

(31)

Note that we do not need to know κ to update our guess. If f (t+1)(i) is not close enough

to f (t)(i), go to step 2. Otherwise, go to the next step.

8. Find λ by∫JwH(j)dj = 1 (the normalization defined in equilibrium),

1 =

∫J

wH(j)dj =

∫J

f(j)1

1−σ dj = λ1

(1−σ)(2σ−1)

∫J

wH(j)1

1−σ dj

So,

λ =[ ∫

J

wH(j)1

1−σ dj](σ−1)(2σ−1)

From here, find wH(i).

9. Find κ,

κ =f(i)∫

JK(j, i)f(j) dj

=f(`)∫

JK(j, `)f(j) dj

36

The above should hold for all i and `. This step, thus, is also a check that the solutions

to integral equations are correct. Then, calculate:

WH = N1θHκ

11−σ

Once wH(i) and WH are known, it is straightforward to calculate all other equilibrium

objects.

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